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RAND Journal of Economics
Vol. 32, No. 4, Winter 2001
pp. 1–27
Competitive price discrimination
Mark Armstrong∗
and
John Vickers∗∗
We model firms as supplying utility directly to consumers. The equilibrium outcome of competition
in utility space depends on the relationship π (u) between profit and average utility per consumer.
Public policy constraints on the “deals” firms may offer affect equilibrium outcomes via their
effect on π(u). From this perspective we examine the profit, utility, and welfare implications of
price discrimination policies in an oligopolistic framework. We also show that an equilibrium
outcome of competitive nonlinear pricing when consumers have private information about their
tastes is for firms to offer efficient two-part tariffs.
1. Introduction
This article has two main aims. First, we propose a framework that we hope will be useful
in analyzing situations where firms compete by offering various “deals” involving tariffs, bundles
of outputs, and so on. Models of markets where firms simultaneously compete on a number of
dimensions are often hard to work with. In many cases, however, this analysis can be simplified
by taking a dual approach and modelling firms as competing directly in utility space. Viewed
from this perspective, a firm’s strategy is just to choose the (scalar) level of utility, or “value for
money,” offered to customers. A firm’s market share naturally increases the more utility it offers
relative to its rivals. Central to the analysis of competition in utility space is the function π (u),
which gives the maximum profit per consumer a firm can extract when it provides utility u to its
consumers. This function succinctly captures not just the relevant cost and demand information;
it also reflects any public policy (or other) constraints on the deals firms may offer. The second
aim of the article is to illustrate the usefulness of this dual approach by examining the profit,
utility, and welfare implications of public policy toward various kinds of price discrimination in
oligopoly.1
∗
Nuffield College, Oxford; mark.armstrong@nuffield.oxford.ac.uk.
All Souls College, Oxford; john.vickers@economics.oxford.ac.uk.
We are grateful for comments from Simon Anderson, Frank Fisher, Anke Kessler, Preston McAfee, John Moore,
Babu Nahata, Barry Nalebuff, and Dick Schmalensee, and we especially thank Lars Stole and two referees for very
detailed advice that improved the clarity and accuracy of this article. All remaining errors are our own.
1 The modelling of firms as competing in utility space is also a useful way to think about competition in network
industries. For instance, see Armstrong (forthcoming) for a survey of recent contributions to this topic, viewed from the
perspective of firms offering utility directly to their subscribers.
∗∗
Copyright © 2001, RAND.
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Competition in utility space. We suppose that consumers differ in their tastes for the
utilities supplied by the firms (although the framework continues to be valid in the extreme case
of perfect substitutes). We do this by using a discrete-choice framework, so that if u i is the utility
supplied by firm i, a consumer’s net utility is u i plus an idiosyncratic shock. (Thus we can think
of u i as the “average” utility offered by firm i to its population of consumers.) This framework
includes such familiar models of product differentiation as the Hotelling location model and
the logit model. As well as being convenient to work with from a theoretical perspective, such
a framework accords well with the simplest forms of many econometric models of consumer
choice.
A convenient feature of this approach is that the effect of the profit function in equilibrium is
completely separable from the effect of the discrete-choice model. For instance, in a symmetric
model with a fixed number of firms, we show that equilibrium utility maximizes an additively
separable function
φ(u) + log π (u),
where φ is an increasing function depending only on the discrete-choice framework (see Proposition 1 below). Similarly, in a symmetric circular Hotelling model with free entry, we show that
equilibrium utility maximizes
u
1
−
,
τ2
π (u)
where τ is a parameter measuring the competitiveness of the market. Thus one can perform comparative statics analysis on the profit function and on the discrete-choice framework independently,
and in the article we do this repeatedly. This analysis is contained in Section 2.
Perhaps the article closest to our competition-in-utility-space approach is Bliss’s (1988)
model of retail competition, in which the role of a shop is to assemble a collection of goods in one
place and offer a price list to consumers. The value of a price list to a consumer is measured by an
indirect utility function. A recurrent theme in Bliss’s analysis is that retailers choose Ramsey-like
markups. This is because a firm’s problem can often be decomposed into (i) how much utility to
offer to consumers, and (ii) how to set prices to maximize profit while offering a given consumer
utility. Whereas Bliss focused on (ii), which is a Ramsey problem, our main concern is with (i),
though (ii) is implicit in our analysis.
While our competition-in-utility-space method is perhaps less familiar, we embed this approach into the well-known discrete-choice framework for consumer choice—see Anderson, de
Palma, and Thisse (1992) for a description of discrete-choice models, and Caplin and Nalebuff
(1991) for a general model of imperfect competition that includes the discrete-choice model as
a special case. Although there are exceptions, most discrete-choice models involve, first, an assumption that product characteristics are exogenously fixed before competition takes place, and,
second, that consumers have unit demands. In particular, a firm’s only decision is the price at
which it offers its single product, while a consumer’s only decision is which firm (if any) to buy
from. The latter assumption is too strong for many applications of interest—for instance, situations where firms offer multiple products and where consumers may wish to purchase multiple
units of a given product—and one aim of the article is to extend the discrete-choice framework
to cover multiproduct, multiunit situations.
Caplin and Nalebuff (1991) and Anderson, de Palma, and Nesterov (1995) provide an earlier
analysis of elastic consumer demand, though the focus is still on single-product firms. These
analyses make the simplifying assumption that consumers make all their purchases from a single
firm, and we shall follow the same approach here.2 This “one-stop-shopping” assumption is not
innocuous, and in future work it would be useful to analyze the implications of allowing consumers
2 Rochet and Stole (forthcoming) have consumers choosing qualities rather than quantities, although formally the
analysis is similar. However, the interpretation of the choice as quality—so that consumers have unit demands—has the
benefit that there is no need to make an assumption that consumers purchase from a single firm.
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to pick and choose from the various products offered by firms to make up their own bundle. It
is certainly true in much of our analysis that in equilibrium, consumers will choose to obtain all
products from a single firm, since that way they incur the transport cost only once (in the Hotelling
context) and pay the fixed charge of a two-part tariff only once (when such tariffs are offered).
However, it does not follow from this that the ability of consumers to shop from two firms has no
effect on the equilibrium. (This issue is discussed further in Section 4.)
Once we allow for elastic consumer demand, we have to decide how the additive shock that
underpins the discrete-choice model interacts with a consumer’s volume of demand. For instance,
in a Hotelling transport cost model, one could imagine incurring the transport cost per unit of
product consumed, or, alternatively, the cost could just be incurred per trip, independently of the
volume of consumption. Because it greatly simplifies the analysis, we take the latter approach.3
However, our model of price discrimination in Section 4 could be interpreted as one in which
consumers pay a transport cost per product (but not per unit of each product).4
Competitive price discrimination. The second main aim of the article is to use this
competition-in-utility-space framework to investigate the profit, consumer surplus, and welfare
implications of various price discrimination policies in oligopoly. We define a firm’s pricing to
be discriminatory if it sells units of its output(s) at different prices when marginal costs do not
differ correspondingly. We will see that the welfare effects of permitting price discrimination
in oligopoly depend in large part on whether discrimination takes the form of what we term
intrapersonal discrimination or whether it is interpersonal discrimination. To illustrate the distinction, consider the case of air travel. Different kinds of tickets, say “business” or “leisure”
tickets, can be charged at different rates, depending on the consumer demand characteristics. It
may be that travellers are segmented, so that some people always want one kind of ticket while
others always want the other. When these tickets have different prices we would call this practice
interpersonal price discrimination, since different consumers face different prices. Alternatively,
it could be that a given traveller demands a mix of business and leisure tickets, depending on the
nature of the planned journeys, and differential charging here is an example of intrapersonal price
discrimination.5
We analyze three information structures:
Homogeneous case. In this case, consumers are assumed to be “homogeneous,” in the sense
that all consumers generate the same level of profits for a given level of utility, i.e., that the
function π(u) is the same for all consumers. (This is a slight abuse of terminology, however, since
consumers continue to differ in their tastes for the utilities offered by firms.)
Observable heterogeneity case. Here consumers are heterogeneous and differ by an observable
characteristic on which firms can base their tariffs. Examples might be where consumers are
regionally segmented, or where the age or occupation of the consumer can be determined.
Unobservable heterogeneity case. Consumers are heterogeneous and differ by unobservable characteristics (in addition to their differing tastes for the utilities offered by firms). Thus the profit
function π (u | θ ) here depends on a parameter θ that is private information to consumers.
In Section 3 we discuss in more detail the simple case of homogeneous consumers who
make all their purchases from a single firm, and here we can apply our competition-in-utilityspace framework directly. Clearly, in the homogeneous consumer case, the only possible kind of
3 The same assumption is made in Anderson, de Palma, and Nesterov (1995). Similarly, in their model of quality
choice, Rochet and Stole (forthcoming) assume that the transport cost does not depend upon a consumer’s choice of
quality.
4 See Stole (1995), and the references therein, for an analysis of the more complicated case where transport costs
(or brand preferences) do interact with the quantity/quality choice.
5 It is fair to say that many economists would reserve the term “price discrimination” for what we call interpersonal
price discrimination. For instance, Varian (1989) defines (third-degree) discrimination to be where different consumers
are charged different prices.
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price discrimination is of the intrapersonal variety. The key idea is that a more constrained firm
will be able to make less profit for a given utility level than would a less constrained firm. Thus,
the function π D (u) if a firm practices price discrimination of some kind will generally differ from
the function π N (u) if it cannot: indeed, π D (u) ≥ π N (u) with strict inequality almost always. Both
nonlinear pricing—which, in the homogeneous case, reduces to two-part tariffs—and third-degree
forms of discrimination are discussed, although the analysis is more general than this and covers
virtually any constraints placed on firms in the way they can offer deals to their customers.
When there are fixed costs associated with serving consumers, so that marginal-cost pricing
of services does not allow firms to break even, greater flexibility allows firms to offer a higher
level of utility before losses are made. In such cases, allowing firms in competitive markets greater
flexibility causes an increase in consumer utility and overall welfare (Proposition 2). If there is no
such fixed cost, then the analysis required is more delicate, and we show that greater flexibility
causes profits and overall welfare to increase in competitive markets, although now consumers
may be made worse off (Proposition 3). The results for welfare are true both when there is a fixed
number of firms and when there is free entry. The key feature of the homogeneous consumer
framework, combined with the one-stop-shopping assumption, is that firms, when permitted to
do so, choose socially desirable prices—i.e., profits are maximized subject to a consumer utility
constraint. Thus, firms will make their prices relate to the underlying costs and to the demand
elasticities in the correct Ramsey manner. In particular, (i) when two-part tariffs are used then
marginal prices equal marginal costs, and (ii) when linear pricing is used, then prices are, roughly
speaking, higher for those products with less elastic demand.
Section 4 covers cases where consumers are heterogeneous, and we analyze both observable
and unobservable heterogeneity. The focus here is naturally on interpersonal price discrimination. We cannot directly apply the competition-in-utility-space framework here, although various
features of that model will be useful ingredients in the analysis. In the first part of Section 4
we discuss the case of interpersonal third-degree discrimination, and in contrast with the homogeneous consumer case, the welfare consequences of competitive price discrimination are
ambiguous. For instance, it may be that competition forces firms to set a lower price in less elastic
markets compared to the no-discrimination case. More generally, it will only be by chance that
prices are Ramsey-like. There is thus a fundamental difference between the “intrapersonal” and
“interpersonal” forms of discrimination in terms of their welfare effects. But although the total
welfare effect is ambiguous, the effect on consumers and profits separately is not: profits increase
but aggregate consumer utility decreases if discrimination is permitted in competitive markets
(Proposition 4).
In the second part of Section 4 we discuss a model with unobserved heterogeneity, where
consumers have private, product-specific information about their tastes (in addition to their private
information about their tastes for the deals offered by firms). We make two important assumptions.
First, we assume that the market is competitive, in the sense that all consumers choose to participate
in the market in equilibrium. Second, we assume that the “vertical” taste parameter does not interact
with the “horizontal” taste parameter. (This assumption has two parts: the two taste parameters
are statistically independently distributed, and, as discussed above, the horizontal taste parameter
is unaffected by the quantities chosen by consumers.) Under these conditions it is an equilibrium
for firms to offer cost-based two-part tariffs (Proposition 5). In particular, there is no screening in
this equilibrium, and all consumers face the same marginal prices. In effect, the need to compete
dominates any benefit derived from screening consumers. Thus, just as in the homogeneous
consumer case, allowing firms to have maximum flexibility in tariff choice can induce an efficient
outcome.
Contrasts with monopoly price discrimination. The above results contrast with familiar
results for monopoly; see Varian (1989) for an analysis of price discrimination with monopoly. A
general disadvantage of price discrimination—both in monopoly and in oligopoly—is that output
is suboptimally distributed to consumers because their marginal utilities will be unequal. This
drawback may be overcome if price discrimination leads to a sufficient increase in output. We
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can deduce that if output falls when discrimination is introduced, then welfare must also fall.
(When we discuss interpersonal third-degree discrimination in duopoly, we shall use this fact
to find conditions under which discrimination is socially undesirable.) For instance, in the case
of third-degree price discrimination, with independent linear demands (and all markets served),
total output is exactly the same with discrimination as without, so discrimination reduces welfare. More generally, the sign of the total output effect in the monopoly case can be related
to the convexity/concavity properties of demands—see Varian (1989). The monopoly model of
price discrimination does not distinguish between the intrapersonal and the interpersonal kinds
of discrimination, since all that matters for the analysis is aggregate demand. As discussed above,
though, we argue that this distinction makes a crucial difference to the desirability of discrimination in oligopoly, and in the homogeneous consumer case we argue that price discrimination
is unambiguously good for welfare in competitive markets. In particular, we can simply infer
without explicit analysis that the output effect is positive if discrimination is permitted.
Turning next to price discrimination when there is unobservable heterogeneity, Armstrong
(1996) and Rochet and Choné (1998) analyze multiproduct nonlinear pricing by a monopolist
facing consumers with private information about their tastes. It is argued that the problem of
designing the profit-maximizing nonlinear tariff is complex, and explicit solutions seem obtainable
in only a few cases. In particular, Armstrong shows that a monopolist will generally choose to
exclude some consumers from the market at the optimum, while Rochet and Choné show that
for those consumers who do participate there will be “bunching,” i.e., that some consumers with
different tastes will choose to purchase the same bundle of products. In practice, these two features
of the optimum make it hard to find closed-form solutions. By contrast, in Section 4 we shall
show that these same models often have a very straightforward equilibrium—without exclusion
or bunching—when there is a reasonable degree of competition.
Other work on price discrimination in oligopoly. The literature on price discrimination
in competitive settings is not as extensive as the analysis for monopoly, though see Varian (1989)
and Wilson (1993) for some references. One very relevant article, however, is Holmes (1989),
who examines the output and profit effects of (interpersonal) third-degree price discrimination
in a symmetric duopoly model with product differentiation. (We examine a very similar model
in Section 4.) He shows that the sign of the total output effect depends not only on the convexity/concavity properties of industry demands—as in the monopoly case—but also upon a term
involving interfirm cross-price elasticities, which reflect competitive interactions. Holmes also
shows that it is possible for equilibrium profit to be higher when price discrimination is banned.
However, few clear-cut results are derived, and little is said about the welfare effects of allowing
discrimination. In Section 4, however, we shall argue that as markets become very competitive
the Holmes model becomes easier to interpret and stronger results emerge; we show that the
effect on profits and on aggregate consumer welfare is unambiguous, and we provide a simple
criterion for predicting when the total output effect is negative (which implies that discrimination
is undesirable).6
Turning to nonlinear pricing in oligopoly, Stole (1995) considers—as we shall—nonlinear
pricing in oligopoly when firms are spatially differentiated. Consumers differ both in their brand
preferences (horizontal differentiation) and their quality preferences (vertical differentiation).
He characterizes optimal pricing schedules and shows how they depend on consumers’ private
information about brand and quality preferences, and on the intensity of competition. A major
difference between Stole’s model and ours is that he allows the two kinds of private information to
interact, in that transport costs are assumed to depend on the quantities consumed (and hence on a
consumer’s taste for the output), whereas we make the simplifying assumption that transport costs
are a lump-sum cost. A model very similar to ours is Rochet and Stole (forthcoming), which was
prepared independently of this work. They assume the same lump-sum transport cost technology
as we do, and find conditions implying that efficient two-part tariffs emerge as an equilibrium.
6
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Customer poaching. Finally, we mention one important kind of price discrimination that
our framework cannot easily handle, when a firm offers a deal to those customers who “belong”
(in some sense) to rival firms that differs from the deal it offers its “own” customers. In most
cases we expect the firm to offer a better deal to rival firms’ customers in order to build market
share. A feature of such models is that, compared to the situation where deals are uniform,
price discrimination can act to intensify competition, in the sense that all prices come down and
equilibrium profits decrease.7 (This is in direct contrast with, say, our model in the first part of
Section 4, where allowing price discrimination causes some prices to rise and some to fall, while
total profits increase.) The basic reason why this kind of discrimination intensifies competition
across the board is that it makes it cheaper for a firm to target its rivals’ customers without
damaging the profits it can extract from its own customer base: when all firms target each others’
customers, though, all prices come down.8
In our model there is no sense in which customers “belong” in any observable way to one
or another firm. However, the kinds of situations where customers are associated with particular
firms include the following: First, in a spatial model Thisse and Vives (1988) suppose that a
consumer’s location is observable, so that consumers are known to be more loyal to one firm
than the other. In this case, when firms can base their prices on consumer location, all prices
might fall compared to the case where prices are constrained to be uniform. Second, Corts (1998)
examines a market where one firm supplies a higher-quality product than the other. There are two
identifiably different consumer groups, one that cares about quality and one that cares only about
price. Corts shows that allowing discrimination could cause both prices to fall compared to the
uniform price regime. The reason is that the two firms behave differently in response to the ability
to discriminate: the high-quality firm will choose to set the higher price in the “choosy” market,
whereas the low-quality firm sets its low price in that market, and so competition is stronger in
both markets. Finally, there is the interesting recent literature on customer poaching, including
Villas-Boas (1999) and Fudenberg and Tirole (2000). Here, the focus is on the dynamics of market
share. Firms know whether a consumer was a previous customer and so can target other consumers
accordingly. Again, a result is that all prices might fall with this kind of discrimination.9
2. Competition in utility space
A framework. There are two symmetrically placed firms, A and B, facing differentiated
consumers. (This analysis applies equally to any fixed number of firms—see below for the
extension to endogenous numbers of firms.10 ) Consumers buy all products from one or the other
firm, or they consume an outside option. Suppose that the “deal” offered by firm i gives each
consumer a utility u i , and that the outside option gives utility u 0 . We use a discrete-choice
framework and assume that a consumer of type ζ = (ζ0 , ζ A , ζ B ) has net utility u i + ζi if he buys
from firm i, and net utility u 0 + ζ0 if no purchase is made. Therefore, a consumer’s maximum
utility is max{u A + ζ A , u B + ζ B , u 0 + ζ0 }. The triplet ζ = (ζ0 , ζ A , ζ B ) is distributed across the
population according to a known distribution. There is a continuum of potential consumers, with
measure normalized to one.
Aggregate consumer utility is given by
V (u A , u B ) ≡ E max{u A + ζ A , u B + ζ B , u 0 + ζ0 } ,
7 In fact, Holmes (1989) shows that the same effect can occur when consumers do not “belong” to particular firms.
Moreover, Nahata, Ostaszewski, and Sahoo (1990) show that when profit functions are not single-peaked, the same effect
can occur even in monopoly.
8 Armstrong and Vickers (1993) present a model where price discrimination can cause all prices to rise. An
incumbent multimarket firm faces a threat of entry in one market. Permitting discrimination makes the incumbent react
aggressively to entry—since it does not have to sacrifice profits in the captive market—which can deter entry.
9 For instance, Villas-Boas has infinitely-lived firms facing an overlapping sequence of two-period consumers,
and he shows that in the steady state, prices can be lower when there is “customer recognition” than without.
10 The only result that depends on duopoly seems to be the fact that we can rule out asymmetric equilibria in
Proposition 1 below. See Perloff and Salop (1985) for a discussion of the issue.
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where E denotes expectation with respect to the distribution for ζ . By the envelope theorem the
function V has partial derivative11
V1 (u A , u B ) ≡ s(u A , u B ),
where s(u A , u B ) is the number of consumers who choose to buy from firm A given the pair of
gross utilities u A and u B . (Differentiating V with respect to u A has two effects: (i) the direct
effect on those consumers who buy from A anyway, which is just s(u A , u B ), and (ii) a shift in
the set of consumers who choose to buy from A. However, those consumers on the boundary in
(ζ0 , ζ A , ζ B )–space between buying from A and other options are indifferent between A and the
other options, so there is no aggregate consumer welfare effect from (ii) at the margin.)
We focus throughout on the case where consumers’ tastes for the two firms’ products are
symmetrically distributed, so that V (u A , u B ) ≡ V (u B , u A ). This implies that the number of B’s
consumers is just s(u B , u A ). Clearly it is necessary that s(u A , u B ) + s(u B , u A ) ≤ 1, with equality
if all consumers participate. It is also necessary that s is increasing in the first argument and
decreasing in the second.12
Suppose that if a firm offers consumers a “deal”—i.e., package of products for sale using
a certain tariff—that gives them utility u, then its maximum profit per consumer is π (u). For
this function to be well defined, we assume the firms have constant-returns-to-scale technology
in serving consumers, so that profit per consumer does not depend on the number of consumers
served. Until we reach Section 4, we assume consumers are homogeneous in the sense that the
function π(u) is the same for all consumers. The form the relationship π (u) takes depends on the
cost structure of the firms, on the consumer demand functions they face, and also on the types of
tariff they are permitted to offer—see the next subsection for examples. Using this notation, firm
i = A, B wishes to maximize s(u i , u j )π (u i ) given the rival’s choice of utility u j .
We need to make some regularity assumptions in order to guarantee existence of equilibrium.
First, we impose the following condition on s:13
s1 (u A , u B )
is weakly increasing in u B .
s(u A , u B )
(1)
Since s is decreasing in the second argument, this condition requires that s1 not be decreasing
too fast in the second argument. In economic terms, this implies that firms’ reaction functions in
utility space are upward sloping. Second, we need to ensure that very low utility levels ultimately
drive all consumers away. To be precise, we assume that the “collusive” utility level u ∗ , which
is defined to be the utility that maximizes (symmetric) joint profits, exists. More specifically, we
assume
there exists u ∗ > −∞ that maximizes s(u, u)π (u).
(2)
Finally, it is economically natural to assume that π (u) is ultimately negative for large values of u,
so let ū be the largest utility level that allows a firm to break even. (For instance, supplying very
high utility ultimately involves offering products at prices below costs.) Therefore, assume there
exists ū defined by
π (ū) = 0, π (u) < 0 if u > ū.
(3)
11 We assume that this derivative exists, which requires that there is only a measure-zero set of potential consumers
who can be indifferent between any pair of options. A sufficient condition for this is for ζ to be distributed according to
a bounded density function.
12 To see this algebraically, one can show, for example, that the set of consumers who buy from A contracts if u
B
is increased.
13 For s to be differentiable, we can assume that the density function for ζ is continuous. In fact, the Hotelling
example we often use has kinked market-share functions. However, when all consumers participate in the market—as is
the case below—this problem does not arise.
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In this framework a symmetric Nash equilibrium in utility space exists, the equilibrium is
often unique, and, most importantly for our purposes, there is a simple formula for the equilibrium
utility:
Proposition 1. Assume (1)–(3) hold. Let φ(u) be a function satisfying
φ (u) =
s1 (u, u)
≥ 0.
s(u, u)
(4)
(This is unique up to a constant.) Then a symmetric equilibrium is given by each firm choosing
utility û that maximizes
φ(u) + log π (u).
(5)
This equilibrium utility û is strictly greater than the collusive utility level u ∗ . There are no
asymmetric equilibria. This is the unique (pure-strategy) equilibrium if (5) is concave.
Proof. All proofs are in the Appendix.
Thus equilibrium in this market involves the maximization of the sum of log profits and an
increasing function φ of consumer utility. In this result, the term s1 /s in (4) gives the proportional
increase in a firm’s customer base caused by unilaterally raising utility when both firms offer the
same utility level. This is a natural measure of the degree of product substitutability in the market.
The larger the term, the more weight is placed on consumer utility compared to profits in the
equilibrium.14
Examples of profit functions π (u). In this section we list a few examples of π (u) derived
from primitives in the market. In all cases except the last these will be necessary ingredients in
the subsequent analysis of price discrimination. In all cases we assume consumer preferences are
quasi-linear in income.
Single-product linear pricing. Suppose firms offer a single product and each consumer has the
demand function Q(P). Let V (P) be the consumer surplus function associated with Q(P), so
that V (P) ≡ −Q(P). (Recall that preferences are assumed to be quasi-linear.) Therefore, a
consumer’s utility is just u = V (P). Each firm has a constant marginal cost c for providing the
service, together with a fixed cost per consumer for providing service to a consumer of k ≥ 0.15
A firm’s profit per consumer if it offers utility u = V (P) is therefore Q(P)(P − c) − k, so that
π (u) is defined implicitly by
π(V (P)) ≡ Q(P)(P − c) − k.
(6)
If k = 0, then the maximum break-even level of utility, which defines ū in (3), corresponds to
marginal-cost pricing, i.e., ū = V (c). On the other hand, if k > 0, then the maximum break-even
level of utility is the one corresponding to average-cost pricing, i.e., if P av > c is the lowest price
that allows a firm to break even with linear pricing then ū = V (P av ). Differentiating (6) yields
π (u) = −(1 − σ (u)),
(7)
where
σ (V (P)) ≡ (P − c)
−Q (P)
.
Q(P)
14 See Bergstrom and Varian (1985) and Slade (1994) for an analysis of when an oligopoly in equilibrium can be
considered as maximizing a single objective function (as here). More generally, the duopoly game described in Proposition
1 is a “potential game”—see Monderer and Shapley (1996) for details.
15 These per-consumer fixed costs are often relevant, as with the costs of handsets in mobile telephony, the costs of
laying cables into homes in fixed telephony, the cost of digital decoders in pay-TV markets, or even the costs of parking
spaces and checkout staff in shops.
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(The function σ represents the elasticity of demand at a given utility level expressed in terms of
the markup (P − c) instead of price P.) If the weak condition σ (u) ≤ 0 holds when u ≤ V (c),
then (7) implies that π(u) is concave in u when π (u) ≥ 0.16 It will be useful for later to note that
when k = 0, so that ū = V (c), (7) implies that
π (ū) = −1.
(8)
Multiproduct linear pricing. Suppose, generalizing the above, that firms sell n products.
Consumer demands may be interdependent across products, and if p = ( p1 , . . . , pn ) is the vector
of prices, then each consumer’s demand for product i is qi (p). A consumer’s surplus is v(p),
where ∂v/∂ pi ≡ −qi . Firms incur a fixed cost k ≥ 0 of supplying a customer, and a vector of
constant marginal costs c = (c1 , . . . , cn ) for supplying the n products.
In this framework, π(u) is given by the solution to the Ramsey problem:
π(u) = max :
n
p≥0
qi (p)( pi − ci ) − k | v(p) = u .
(9)
i=1
If k = 0, then ū, the maximum utility that allows a firm to break even, is again induced by
marginal-cost pricing, i.e., ū = v(c). On the other hand, if k > 0, then pricing at marginal cost
will not cover costs, and ū will be lower. Indeed, unless at least one product has perfectly inelastic
demand (which we rule out from now on), there will be deadweight losses and ū < v(c) − k. In
sum,
= v(c)
if k = 0
ū
(10)
< v(c) − k if k > 0.
For a given u let λ(u) be the Lagrange multiplier used to solve problem (9).17 For any λ ≤ 1 let
p̂(λ) be the price vector that maximizes the function
n
qi (p)( pi − ci ) + λv(p).
(11)
i=1
In particular, p̂i (1) = ci . The price vector that solves problem (9) is therefore p̂(λ(u)), where λ(u)
is defined implicitly by
v(p̂(λ(u))) ≡ u.
(12)
π (u) = −λ(u).
(13)
The envelope theorem implies that
Note that λ(u) is increasing, so that π(u) is concave. Suppose that k = 0, so that ū = v(c). Then
marginal-cost pricing (which corresponds to λ = 1) is the policy that generates utility ū, i.e.,
18
π (ū) = −λ(ū) = −1.
(14)
Two-part tariffs. Consider the same framework as above but allow firms to use two-part tariffs
instead of linear prices. Suppose a firm chooses a vector of marginal prices p and a fixed charge
16 This discussion is very closely related to Section 3 of Anderson, de Palma, and Nesterov (1995), where the case
of single-product firms competing for consumers with elastic demand is analyzed. In particular, the function η in their
assumption A4 is our function σ in the text, i.e., σ (V (P)) ≡ η(P).
17 We assume demand functions are well behaved enough so that the Lagrangian method is valid—see Diamond
and Mirrlees (1971) for a full discussion of when the Lagrange method works for the Ramsey-unique solution.
18 To see this, note that the optimized value of (11), viewed as a function of λ, is convex in λ and has derivative
equal to v(p(λ)). Therefore, v(p(λ)) is increasing in λ, which, in combination with (12), implies that λ(u) is increasing.
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f . Then consumer utility is u = v(p) − f . The profit-maximizing method of generating u is to
set prices equal to marginal cost, in which case utility is u = v(c) − f . Profit as a function of u is
therefore given by the linear relationship
π (u) = v(c) − k − u,
(15)
and the maximum utility that allows a firm to break even is
ū = v(c) − k.
(16)
Comparing this expression with (10), we see that when there are per-consumer fixed costs, the use
of two-part tariffs enables a higher level of utility to be achieved compared with linear pricing.
Advertising in media markets. Although the remainder of the article is focused on pricing issues,
here we briefly consider how the competition-in-utility-space framework can be applied to the
equilibrium balance between advertising and retail charges in media markets such as newspapers
and pay-TV. Suppose that consumers obtain utility from consuming the media product equal to
u = v − D(a) − p,
where v is the gross surplus from consuming the service, a is the volume of advertising, D(·) is
the function giving how much consumers dislike advertising, and p is the charge to the consumer
for the product. Advertising revenues for a media firm are closely related to their subscriber base,
so suppose that if a firm chooses advertising volume a it obtains advertising revenue R(a) per
subscriber. Suppose that a firm incurs a cost k per subscriber in providing the service. Therefore,
its profit per subscriber is p + R(a) − k, which in terms of u is v − u − k + R(a) − D(a), and so
its maximum profit per subscriber—if the firm is free to choose the balance between consumer
prices and advertising—is given by the function
π(u) = v − u − k + max{R(a) − D(a)},
a
which is of the same linear form as the two-part tariff case above.
Examples of discrete-choice models. Here we provide two examples of preferences that
generate particular instances of the market-share function s(u A , u B ). (See Anderson, de Palma,
and Thisse (1992) for further examples.)
Hotelling preferences. The simplest Hotelling model involves customers being uniformly located
on the unit interval [0, 1]. A customer located at 0 ≤ x ≤ 1 receives net utility u A − t x if she
buys from A and net utility u B − t(1 − x) if she buys from B. Thus firm A is located at zero and
firm B is located at one. The parameter t > 0 is a consumer’s “transport cost” per unit distance
of buying from one firm or the other. If an agent buys from neither firm, she receives zero net
utility.19 Therefore, a consumer at x will buy from firm A if (i) u A − t x ≥ u B − t(1 − x) and (ii)
u A − t x ≥ 0. These inequalities can be rearranged to give
1
uA − uB
uA
x ≤ min :
1+
,
.
2
t
t
Therefore, s is given by
1
uA − uB
uA
s(u A , u B ) = min :
1+
,
.
2
t
t
(17)
19 Thus, in terms of the notation developed above, u = 0, ζ ≡ 0, ζ = −t x, and ζ = −t(1 − x), where x is a
A
B
0
0
random variable uniformly distributed on [0, 1].
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(There are obvious modifications when u A and u B differ by so much that s above lies outside
the interval [0, 1], which never occurs in equilibrium.) Unfortunately, the Hotelling model has
the well-known technical drawback that market shares are kinked at a point where there exists
a consumer who is indifferent between the three options of buying from A, buying from B, or
buying nothing. This means that we cannot directly apply Proposition 1. However, the following
similar result is easily obtained:
Lemma 1. Suppose that log π(u) is concave and that
t ≤ 2β,
(18)
where β > 0 is the value of u that maximizes (1/2) log u + log π (u). Then equilibrium utility,
denoted û(t), maximizes
u
+ log π (u).
(19)
t
Condition (18) implies that if û(t) maximizes (19), then û(t) ≥ β and so all consumers are willing
to pay the transport cost to obtain this utility, i.e., that the market is covered.
For an illustration of Lemma 1, consider the case where two-part tariffs are used, so that the
profit function is given by (15). Then Lemma 1 shows that, provided parameters are such that the
market is covered in equilibrium, which in this case reduces to requiring20
t≤
2
[v(c) − k] ,
3
(20)
π̂ (t) ≡ t,
(21)
in equilibrium we have
so that a firm’s profit per customer is equal to the differentiation parameter t. Although tariffs
are not the focus of our approach in this section, it is useful for later to note that the tariff that
generates this level of profits is
T ∗ (q) = t + k +
n
ci qi .
(22)
i=1
This is a two-part tariff with prices equal to marginal costs, and the margin of the fixed charge
over the fixed cost is the product-differentiation parameter t. In particular, this equilibrium tariff
does not depend on any consumer-demand characteristics, and this feature will play a major role
when we discuss tariff design under asymmetric information at the end of the article.
It will be useful to understand the asymptotic properties of the equilibrium as the market
becomes highly competitive. (Note that Lemma 1 certainly applies when t is close to zero.) As
described above, û(t) denotes equilibrium consumer utility with transport cost t. Similarly, let
π̂ (t) ≡ π(û(t)) be equilibrium profits, and let ŵ(t) ≡ û(t) + π̂ (t) be unweighted welfare (not
including consumers’ transport costs which, except in the next subsection, are identical in all
equilibria, and hence irrelevant for welfare). Then we have the following result:
Lemma 2. Suppose π(u) is at least twice continuously differentiable. In the Hotelling duopoly
model for small t, the equilibrium utility, profits, and welfare are given by the following secondorder Taylor approximations:
û(t) ≈ ū − t +
t2
2
π (ū)
π (ū)
π̂ (t) ≈ −tπ (ū) + t 2 π (ū)
20
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ŵ(t) ≈ ū − t(1 + π (ū)) +
t2
π (ū)
2π (ū) + .
2
π (ū)
When we come to discuss the impact in competitive markets of various price discrimination
policies, we will use these Taylor expansions—which relate outcomes to the slope and curvature
of π at ū—to evaluate a change in the tariff regime in terms of its effect on consumers, firms, and
overall welfare.
Logit preferences. With the normalization u 0 = 0 the logit model with product-differentiation
parameter µ implies that the market-share function takes the form
s(u A , u B ) =
eu A /µ
eu A /µ
.
+ eu B /µ + 1
(A significant difference between the logit and the Hotelling models is that in the former there
are always some consumers who choose not to participate.) Then
s1 (u, u) 1 eu/µ + 1
=
,
s(u, u)
µ 2eu/µ + 1
and so Proposition 1 implies that equilibrium utility û maximizes
log π(u) +
u
1
− log 2eu/µ + 1 .
µ 2
(One can check that (1) is satisfied.) When µ is small, this maximization problem is closely
approximate to maximizing log π(u) + u/(2µ), which is the same problem as (19) when t = 2µ.
In particular, all the asymptotic results for the Hotelling model contained in Lemma 2 above are
valid for the logit model as well if we write t = 2µ.
Free entry. Here we extend the model to allow for free entry of firms. We need to be more
specific about consumer preferences here, and so we use the familiar Salop (1979) extension of
the Hotelling model where consumers of mass one are uniformly distributed around a circle of
circumference one.21 Transport cost is t and there is an overall fixed cost per firm K . Let π(u) be
the same per-consumer profit function as previously. We assume that market equilibrium implies
that firms are equally spaced around the circle, and that integer concerns are not significant.
Analogously to Lemma 1 above, equilibrium u for a given number of firms n satisfies
π (u) π (u)
+
= 0.
n
t
(23)
Given equilibrium utility u, the number of firms n is given by the zero-profit condition
π (u) = n K .
We assume that transport cost t is large relative to the fixed cost K so that free-entry equilibrium
allows a reasonable number of firms to enter the market. Define τ by τ 2 = K t. Then the above
pair of conditions implies that
(π (u))2 + τ 2 π (u) = 0
(24)
21 An alternative approach would be to follow Perloff and Salop (1985) and Anderson, de Palma, and Nesterov
(1995) and model endogenous entry by assuming that a consumer’s utility shocks ζi are independently and identically
distributed across all potential firms.
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defines free-entry equilibrium level of gross utility as a function of τ , denoted û(τ ). Assuming that
log π(u) is concave, this implies that −1/π is also concave, and so (24) implies that equilibrium
û(τ ) maximizes the function
u
1
−
,
τ2
π (u)
which is the free-entry version of (19) above. Clearly û(τ ) is decreasing in τ and tends to ū, the
maximum break-even level of utility, as τ → 0. Notice that, like (21) in the duopoly model, when
firms offer two-part tariffs the expression (24) implies that equilibrium profits, excluding K , are
π(û(τ )) ≡ τ .
Since free entry drives profits to zero, total welfare with parameter τ , denoted ŵ(τ ), is just
net consumer utility, which is utility û(τ ) minus average transport costs, and is given by
ŵ(τ ) = û(τ ) −
t
π (û(τ ))
= û(τ ) +
,
4n
4π (û(τ ))
(25)
where the second equality comes from (23).
The following lemma, which is analogous to Lemma 2 derived in the duopoly case, describes
the asymptotic properties of ŵ:
Lemma 3. The second-order Taylor expansion of ŵ(τ ) about τ = 0 is
ŵ(τ ) ≈ ū −
5τ
τ 2 π (ū)
1
√ .
+
4 −π (ū) 8 (π (ū))2
3. Discrimination with homogeneous consumers
For the remainder of the article we examine the welfare effects of allowing firms more
discretion over their choice of tariffs. In this section we focus on the case of homogeneous
consumers, so that, using the terminology of the Introduction, price discrimination must be of the
intrapersonal variety.
Let π D (u) be the profit function if firms can engage in price discrimination in some way, and
let π N (u) be the corresponding function if no discrimination is possible. Note that the cause of a
change in tariff regime need not be a change in public policy, but could result from technological
developments that facilitate more complex tariffs. To illustrate the following discussion, consider
two natural kinds of price discrimination that use the profit functions derived in Section 2.
Two-part tariffs versus linear prices. Here, π N is the profit function corresponding to
multiproduct linear pricing, as in (9), and the break-even level of utility, ū N , is given by (10).
When two-part tariffs are used, the profit function, π D , is given by (15), and the break-even
level of utility, ū D , is given by (16). We will see that the relative performance of these tariffs in
competitive markets depends on whether or not there is a fixed cost k associated with serving a
customer. Comparing (16) with (10), we see that when k > 0 we have ū D > ū N . In contrast,
when k = 0, we have ū D = ū N = v(c), while (14) and (15) give π D (ū) = π N (ū) = −1.
Intrapersonal third-degree price discrimination. Suppose the multiproduct firms have
the same marginal cost c for each product. A ban on price discrimination requires a firm to set
a uniform (linear) price pi ≡P for all products, in which case the demand function aggregated
n
over all products is Q(P) ≡ i=1
qi (P, . . . , P). Let V (P) ≡ v(P, . . . , P) be consumer surplus
at the uniform price P. Then the single-product linear pricing analysis in Section 2 applies, so
π N , the profit function with the uniform pricing constraint, is given by (6). If k = 0, then ū N , the
maximum utility that allows break-even without price discrimination, is just V (c), and (8) implies
that π N (ū N ) = −1. If k > 0, let P av denote the minimum uniform price that allows a firm to
break even, in which case ū N = V (P av ).
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If firms can offer different prices for different services, then π D (u) is given by (9) above (with
c j ≡ c). As above, if k = 0, then ū D = V (c), and from (14) we see that π D (ū D ) = −1. If k > 0,
then—except for the knife-edge case where uniform pricing happens to be the utility-maximizing
way to cover the fixed cost k—we have ū D > ū N , since the firm could choose to set price P av in
each market, but there is generically a more profitable way to generate utility ū N .
We wish to emphasize, however, that the following argument applies more generally than
to this pair of choices, and virtually any pair of profit functions such that π D corresponds to a
situation where firms have more flexibility over their choice of tariff than is the case with π N
will work. For instance, another case might be where π D corresponds to unrestricted multiproduct
linear pricing, and π N is the function induced when firms are not permitted to set any prices below
the associated marginal cost (i.e., where “loss-leading”is banned).22
When firms can engage in price discrimination, they have more ways to generate a given
level of utility, and so it follows that π D (u) ≥ π N (u), where the inequality will often be strict. In
particular, if ū D and ū N are the respective maximum utilities that allow firms to break even, then
ū D ≥ ū N . The above examples—of linear versus two-part pricing and uniform versus nonuniform
pricing—suggest that the inequality ū D ≥ ū N typically is strict when there are positive fixed
costs of supplying services to a consumer (k > 0), for in that case discrimination provides a more
efficient means with which to cover fixed costs. In sum,
k > 0 =⇒ ū D > ū N .
(26)
The fixed per-consumer cost case is illustrated in Figure 1, which applies to the duopoly
Hotelling model. From (19), the equilibrium utilities, denoted u N and u D in the figure, satisfy
tπi (u i ) + πi (u i ) = 0, the tangent to πi (·) at the point u i intersects the horizontal axis a distance t
to the right of equilibrium utility. In fact, because π D (u) is a straight line with π D (u) ≡ −1, this
diagram illustrates the case of two-part tariffs versus linear pricing—see (15) above. Therefore
equilibrium profits with discrimination in this case are t, as in (21) above. The figure is intended
to show what may happen when the market is not particularly competitive (i.e., t is quite large),
in which case equilibrium utility is lower when discrimination is allowed.
On the other hand, when there are no fixed costs to cover, so that k = 0, then we expect that
ū N = ū D = ū, say. The reason for this is that marginal-cost pricing—which is nondiscriminatory—
is the most efficient way to supply utility to consumers. Moreover, when k = 0, we expect that
π D (ū) = π N (ū) = −1. Again, this makes sense from an economic standpoint: when k = 0,
we know that ū corresponds to marginal-cost pricing, and since total welfare per consumer,
u + π (u), is maximized at marginal-cost pricing, it follows that π (ū) = −1. Finally, note that
since π D (u) ≥ π N (u) and the first two terms in a Taylor expansion about ū are equal for the two
profit functions, i.e., π D (ū) = π N (ū) = 0 and π D (ū) = π N (ū) = −1, it is necessarily the case that
π D (ū) ≥ π N (ū). Except in knife-edge situations, this last inequality will be strict.23 In sum,
k = 0 =⇒ ū D = ū N = ū, π D (ū) = π N (ū) = −1, π D (ū) > π N (ū).
(27)
Unfortunately, we have mostly been able to obtain general results only in the competitive
limit, e.g., as the product-differentiation parameter t or τ is close to zero in the Hotelling models.
Results for less competitive markets have been possible to obtain only by specifying the market
and the kinds of tariffs in more detail. For instance, by specifying a functional form for consumer
demand (such as linear independent demand functions) one can explicitly work out the welfare
comparison between nonuniform versus uniform multiproduct linear pricing over the whole range
22 Alternatively, in the media-markets model above, π could be the profit function if firms are unrestricted, and
D
π N could be the one corresponding to when firms are restricted in some way, for instance in the amount of advertising
they can supply.
23 For instance, in the case of intrapersonal third-degree price discrimination, one can show that π (ū) = π (ū)
D
N
only if all the appropriate elasticities of demand for the n products are equal (when evaluated at marginal-cost pricing).
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FIGURE 1
DUOPOLY MODEL WITH k > 0
of transport costs.24 However, one result that holds outside the competitive limit is the following
application of monotone statics:
Lemma 4. In the duopoly model, suppose that ū N = ū D = ū and log π D (u) − log π N (u) is
decreasing in u for u < ū. Then equilibrium utility is lower when price discrimination is permitted
than when it is not.
Thus we see that if π D /π N decreases in u, so that the proportional benefit of engaging in price
discrimination is higher when less utility is offered to consumers, in equilibrium consumer utility
will fall with discrimination. (Of course, this result says nothing about the effect of discrimination
on overall welfare; we will see below that welfare often rises despite a fall in consumer surplus
when discrimination is introduced.) One application of this result is to the comparison between
linear and two-part pricing.
Corollary 1. Suppose k = 0. In the duopoly model, equilibrium utility is lower with two-part
tariffs than with linear pricing.
For the remainder of this section we focus on the competitive limit. We use the Hotelling
discrete-choice model, in both the duopoly and free-entry frameworks. This is because it is
convenient to use a parameterized family of preferences that can allow services to become closely
substitutable. But since the logit model closely approximates the Hotelling model in competitive
markets—see above—all the following results for competitive markets apply equally with that
discrete-choice example as well. A simple corollary of Lemmas 2 and 3 is that in the fixed
per-consumer cost case of (26), we have consumer utility and total welfare being higher when
discrimination is allowed, provided the market is sufficiently competitive:
Proposition 2 (fixed per-consumer cost case). Suppose ū D > ū N . Then
(i) in the duopoly Hotelling model if there is sufficiently strong competition—t close to
zero—both consumer utility and welfare are higher if price discrimination is permitted;
(ii) in the free-entry Hotelling model if there is sufficiently strong competition—τ close to
zero—welfare (which equals consumer utility) is higher if price discrimination is permitted.
24 This was done in an earlier version of the article, and we showed that allowing discrimination was better for
overall welfare for a large range of transport costs, not just for those near zero.
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Notice that Figure 1 applies to the case where ū D > ū N and where the market is not especially
competitive, when consumers are worse off when discrimination is permitted. However, one can
see from the diagram that as t becomes small we have equilibrium utility u D being greater than
the maximum possible utility without discrimination, ū N , and hence greater than the equilibrium
utility without discrimination.
The intuition for this result is straightforward: in competitive markets firms can attract
consumers only by delivering utility that is close to the maximum possible; restricting the ways
in which firms can deliver utility reduces this maximum utility. In the duopoly model, profits are
approximately zero in the competitive case, so welfare then is also higher with discrimination.25
In the free-entry model, profits are always zero, but in the competitive case, transport costs are
close to zero (either because the per-unit transport cost t is small or because there are many firms
and so the distance travelled is small), and therefore welfare here is also increased by allowing
discrimination. This simple result does not depend on the particular Hotelling specification of
tastes. Any parameterized model in which equilibrium utility tends to the maximum possible
break-even utility as competition becomes more intense will have the same property.26
The analysis required is more delicate in the case where there are no per-consumer costs, as
in (27). However, we have the following corollary of Lemmas 2 and 3:
Proposition 3 (no fixed per-consumer cost case). Suppose ū N = ū D = ū, π N (ū) = π D (ū) = −1
and π D (ū) > π N (ū). Then
(i) in the duopoly Hotelling model if there is sufficiently strong competition—t close to zero—
consumer utility decreases and profits and welfare increase if price discrimination is permitted;
(ii) in the free entry Hotelling model if there is sufficiently strong competition—τ close to
zero—welfare (which equals consumer utility) is higher if price discrimination is permitted.
See Figure 2—which is the same as Figure 1 except that ū N = ū D = ū—for an illustration of
this result in the Hotelling duopoly case. In fact, this figure precisely illustrates the case covered
by Corollary 1 comparing linear pricing with two-part tariffs.
Comparing Propositions 2 and 3, notice that, with the duopoly framework, changing one
apparently minor feature of the model specification—whether or not there exist per-consumer
fixed costs—reverses the ranking of the two tariffs as far as consumers are concerned (though
not for overall welfare). A technical reason for this can be found by looking at Lemma 2. When
ū D > ū N , the impact of allowing discrimination in competitive markets is immediately seen
simply by the limiting value of the functions û and ŵ as t tends to zero, and no information about
the derivatives of the functions π N and π D is needed for the comparison. Thus, only “0th-order”
information is needed for the comparison. When ū D = ū N , however, not only are the limiting
values of π N and π D equal, but so are their first derivatives. (Compare (27) with (26).) This means
that second-order information on these profit functions is required to be able to understand the
impact of price discrimination. Roughly speaking, whenever 0th-order effects are sufficient to
compare the two regimes, everyone is better off with discrimination in the competitive limit; when
we have to move to second-order comparisons, the ranking is reversed for consumers.
Why, when there are no fixed per-consumer costs, are consumers worse off with discrimination in competitive markets? Because second-order effects are involved, it seems hard to get
much direct economic intuition. However, one explanation can be found by looking at Lemma 4.
Taking a second-order expansion of πi (u) about ū, one obtains (for i = N , D)
1
πi (u) ≈ ū − u + (ū − u)2 πi (ū).
2
Since we know that π D (ū) > π N (ū), it follows that locally around ū we have that log π D − log π N
25 In fact, Lemma 2 shows that in competitive markets, equilibrium profits are higher with discrimination whenever
π D (ū D ) < π N (ū N ), which is certainly true in the two-part tariff versus linear-pricing comparison.
26 See Perloff and Salop (1985) for a fairly general analysis of when this happens with unit demands.
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FIGURE 2
DUOPOLY MODEL WITH k = 0
is decreasing in u, and we can apply Lemma 4. (In the competitive limit we know that we can
restrict attention to u ≈ ū.)
Finally, we mentioned in the Introduction that one concern of the previous literature has
been to determine the effect of price discrimination on total output. In all the many cases where
we have seen that discrimination raises welfare, we can simply infer that total output rises when
discrimination is allowed. Indeed, it must rise by enough to offset the welfare cost of unequal
marginal utilities.
4. Discrimination with heterogeneous consumers
Interpersonal third-degree price discrimination. In this section we discuss a model first
analyzed by Holmes (1989), in which two firms operate in two distinct markets, denoted 1 and
2, and each supplies a single product in each market. For simplicity, we analyze the duopoly
Hotelling model as before. Firms have the same constant marginal cost c in each market, and
there are no per-consumer fixed costs. Firms offer linear tariffs, and let pi denote a firm’s price
in market i. Suppose the fraction of customers in market 1 is α (and the fraction in market
2 is 1 − α). Each customer in market i has the demand function qi ( pi ). The two markets are
separate, and a customer in one market is not able to purchase the product in the other market. Let
q̄( p) ≡ αq1 ( p) + (1 − α)q2 ( p) denote the average demand across the two markets with common
price p. Let vi ( pi ) denote the consumer-surplus function associated with demand function qi , and
let v̄( p) be that associated with q̄. Let πi (u) be the per-customer profit function in market i, so
that πi (vi ( pi )) = qi ( pi )( pi − c) as in (6). Let ηi = −cqi (c)/qi (c) be the elasticity of demand in
market i with marginal-cost pricing. Without loss of generality, suppose that markets are labelled
so that market 1 is the more elastic:
η1 ≥ η2 .
(28)
We allow the product-differentiation parameter (or transport cost) to differ in the two markets, so
let t be the parameter in market 1 and γ t be that in market 2. In sum, if firms A and B set the
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prices p1A and p1B respectively in market 1, then firm i obtains total profits of
αq1 ( p1i )( p1i
j
1 v1 ( p1i ) − v1 ( p1 )
− c)
+
2
2t
(29)
in that market (and similarly for market 2).
The regimes of no discrimination and discrimination are described below.
Price discrimination. In this case we can consider the two markets separately. From (8) we know
that πi (vi (c)) = −1, and so Lemma 2 implies that equilibrium utilities in the two markets are, to
first order in t, approximately given by
u 1 ≈ v1 (c) − t; u 2 ≈ v2 (c) − γ t.
(30)
Therefore, aggregate consumer welfare is approximately given by
u D (t) ≈ v̄(c) − t [α + γ (1 − α)] .
(31)
From (30) the prices in the two markets are approximately given by
p1 ≈ c +
t
γt
; p2 ≈ c +
,
q1 (c)
q2 (c)
and so market 2 is the more “competitive,” in the sense that equilibrium price is lower there if
price discrimination is permitted, provided that
q2 (c) > γ q1 (c).
(32)
Notice that this condition governing which market has the lower price has nothing to do with the
industry-level demand elasticities that are relevant for welfare: a market could be highly inelastic
at the industry level and yet competition could force firms to choose the lower price there. In more
detail, from (29) a firm’s own-price elasticity in market 1 when the two firms choose the same
price p1 is
p1 q 1 ( p1 )
,
t
where η1 ( p1 ) = − p1 q1 ( p1 )/q1 ( p1 ) is the industry-level elasticity. Thus a firm’s own-price
elasticity in a market, which is inversely related to the equilibrium price, does depend on the
industry-level elasticity η1 , but in highly competitive markets (t close to zero) this factor is
swamped by the second competitive interaction term, which yields the relation (32).
From Lemma 2, aggregate profits are approximately given by
η1 ( p 1 ) +
π D (t) ≈ t [α + γ (1 − α)] .
(33)
Finally, total output under discrimination is approximately given by
Q D (t) ≈ q̄ −
t
[αη1 + γ (1 − α)η2 ] ,
c
(34)
where q̄ = q̄(c).
No price discrimination. A ban on price discrimination requires a firm to set p1 = p2 = p. Using
the expression (29), and the corresponding expression for market 2, the first-order condition for
the equilibrium uniform price p is
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1
d
d
αq1 ( p)π1 (v1 ( p)) + (1 − α)q2 ( p)π2 (v2 ( p)) = t α π1 (v1 ( p)) + (1 − α) π2 (v2 ( p))
γ
dp
dp
or
φ( p)( p − c) = t q̄( p) + q̄ ( p)( p − c) ,
where
φ( p) ≡ α (q1 ( p))2 +
1−α
(q2 ( p))2 .
γ
Writing p = p(t), differentiating this first-order condition gives
d
d q̄( p) + q̄ ( p)( p − c) p (t) = q̄( p) + q̄ ( p)( p − c),
[φ( p)( p − c)] − t
dp
dp
which implies that p (0) = q̄/φ, where these quantities are evaluated at marginal-cost prices.
Therefore, aggregate consumer surplus for small t is approximately
u N (t) = v̄( p(t)) ≈ v̄(c) − t
q̄ 2
.
φ
(35)
Similarly, aggregate profits are approximately given by
π N (t) ≈ t
q̄ 2
.
φ
(36)
Finally, total output without discrimination is approximately
q̄
Q N (t) ≈ q̄ + t q̄ .
φ
(37)
Proposition 4. For sufficiently small t,
(i) except in the knife-edge case where q2 (c) = γ q1 (c), aggregate consumer surplus is lower
and profits are higher if price discrimination is permitted;
(ii) if (32) holds, then welfare is lower when price discrimination is permitted.
Notice in particular that profits always increase if discrimination is permitted in competitive
markets, which shows that Holmes’s (1989) result that profits may fall with discrimination requires
markets to be reasonably uncompetitive. Also, it is aggregate consumer surplus that falls with
discrimination: clearly consumers in the “weak” market benefit from the lower prices caused by
allowing discrimination.27
Figure 3 illustrates this result. The point ( p, p) is the equilibrium uniform price vector, and
iso-welfare, iso-total output, iso-profit, and iso-consumer surplus contours are drawn through this
point. The iso-welfare and iso-output contours are tangent at this point, since welfare is maximized
for a given level of total output when prices are chosen to be uniform. (One can also check that
the former lies inside the latter.) Since we assume that market 1 is more elastic, the iso-profit and
iso-consumer surplus contours are less steep than the iso-welfare contour through this point, with
the former being less steep than the latter. Improvements to consumer surplus and welfare that
maintain the same level of profitability all lie in the region p2 > p1 . However, assumption (32)
implies that firms choose a lower price in market 2 if they are allowed to discriminate, something
27 It is straightforward to show that the equilibrium uniform price lies between the two discriminatory prices. As
discussed in the Introduction, this feature contrasts with models of “customer poaching” in which all prices might fall
with discrimination.
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FIGURE 3
INTERPERSONAL THIRD-DEGREE PRICE DISCRIMINATION
that increases profits but reduces consumer surplus and overall welfare—this is a point like A on
the figure.
Intuitively, the fact that firms reduce the price in the inelastic market and raise the price in
the elastic market causes total output to fall—this is shown rigorously in the proof—and this must
reduce welfare compared to the no-discrimination case. In sum, and in contrast to our earlier
analysis with homogeneous consumers, a firm’s incentives when choosing its pattern of relative
prices may diverge from those of the economy as a whole, and Ramsey-like prices are offered
only by chance.
If the inequality (32) is reversed but not too much, then firms choose to raise price in the
“correct” market, but this effect is not sufficient to cause total output to rise, and the welfare
effect is still negative—this is a point like C on the figure. On the other hand, if (32) is reversed
sufficiently strongly, then the welfare effect is reversed, as in point B on the figure.
Inter- or intrapersonal price discrimination? In this section we have chosen to interpret the
industry as one where there are two disjoint groups of consumers.28 However, to bring out
connections with the analysis of intrapersonal price discrimination in Section 3, we can also
interpret the model as one where there is a single group of “homogeneous” consumers, each of
whom buys two products.
To see this, suppose that a consumer incurs the “transport cost” of going to one firm or the
other on a per-product basis (rather than the per-trip basis of Section 3). Specifically, suppose
that the transport cost is t for product 1 and γ t for product 2 (per unit of distance). Suppose a
consumer lies at location x1 ∈ [0, 1] with respect to product 1, and lies at location x2 ∈ [0, 1]
with respect to product 2, and that the marginal distribution of each xi is uniform on [0, 1]. A
consumer’s demand for product i with the price pi is (1/2)qi ( pi ), and there are no cross-price
effects across the two markets. Then for any pair of price vectors offered by the two firms, it is
clear that the outcomes of this homogeneous consumer model coincide precisely with those of
the segmented market model (setting α = 1/2) discussed earlier in this section: because transport
28
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costs are incurred per product and there are no cross-price effects, the prices offered in one market
do not affect a consumer’s preferences over the choice of firm in the other market. For instance, in
this homogeneous-consumer framework, the profits generated in market 1 for firm i do not depend
at all on firm behavior in market 2 and are given by the expression (29) above (with α = 1/2).
The extent of correlation between a consumer’s location parameters x1 and x2 , i.e., whether or
not consumers systematically favor one firm or the other for the two products, plays no role in
this analysis.
In particular, when transport costs are levied per product rather than per trip, there is no
reason to suppose that a firm will offer Ramsey-like prices for its two products when price
discrimination is permitted, by contrast with the analysis in Section 3. This discussion suggests that
the welfare effects of discrimination are not determined so much by the “homogeneous-” versus
“heterogeneous-”consumer dichotomy that we have so far emphasized; rather, it is determined by
whether the “one-stop-shopping” assumption we imposed in Section 3 is appropriate. If consumers
choose products on a mix-and-match basis from the two firms, then these consumers behave much
as if there were two segmented consumer groups, each of which buys only one product. In
particular, while price discrimination is desirable in competitive markets where one-stop shopping
is the relevant model, when mix-and-match shopping is the more appropriate framework the
welfare effects of price discrimination are less clear-cut.
Finally, it should be emphasized that this close parallel between multiproduct consumers and
segmented, single-product consumers only applies to this setting with linear pricing. For instance,
even with per-product transport costs it will most likely be in the interests of firms to bundle the
two products together when they face multiproduct consumers, so that joint consumption is
encouraged (or perhaps discouraged) endogenously.29 However, this bundling policy makes no
sense when the industry is made up of two segmented groups of single-product consumers.
Price discrimination with unobserved consumer heterogeneity. Here, suppose consumers have private information about their tastes for the services (and not just about their
location). As in Section 2, each firm offersn products, and if it supplies quantities q = (q1 , . . . , qn )
n
ci qi .
to a consumer a firm incurs the cost k + i=1
We employ a similar utility specification as in the monopoly model of Armstrong (1996),
but extended to a duopoly Hotelling model with transport cost t. Specifically, a consumer’s type
is an m + 1 vector (θ , x), where θ = (θ1 , . . . , θm ) is a vector of preference parameters for the n
products and, as earlier, 0 ≤ x ≤ 1 denotes the location of the consumer between the firms. As
before, suppose that consumer location x is uniformly distributed on the interval [0, 1], although
this is not important for the argument. A consumer of type θ has utility, excluding transport costs,
given by
U (θ , q) − T
if she consumes quantities q in return for the payment T . Unlike Armstrong (1996), we need make
no assumptions about the form of the utility function U (e.g., concerning single crossing), nor
about the distribution of θ (such as whether θ is continuously distributed or about independence
of θ1 , . . . , θm ). Finally, let
v(θ ) ≡ max : U (θ , q) −
q
n
ci qi
i=1
be the type-θ consumer’s utility, excluding transport costs, when there is marginal-cost pricing.
Suppose firm j = A, B offers the multiproduct nonlinear tariff T j (q). Let
u j (θ ) = max : {U (θ , q) − T j (q)}
q
29
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See Matutes and Regibeau (1992) for a model along these lines when the two products are perfect complements.
2001.
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be the type-θ consumer’s maximum utility if she buys from firm j, excluding transport costs.
Then the type-(θ , x) consumer obtains net utility u A (θ ) − t x if she buys from A, u B (θ ) − t(1 − x)
if she buys from B, and zero utility if she buys from neither.
We then have the following result:
Proposition 5. Suppose that θ and x are independently distributed, and x is uniformly distributed
on [0, 1]. Provided
2
[v(θ ) − k] for all θ ,
3
each firm offering the cost-based two-part tariff (22) is an equilibrium.
t≤
(38)
Condition (38) is just the condition that ensures all consumers participate with this choice of tariff.
(It is the generalization of (20) to the case of heterogeneous consumers.) We do not make any
claims about the uniqueness of this equilibrium. Indeed, in the general multiproduct framework
used here it does not seem feasible even to calculate A’s best response given a tariff chosen by B
unless B’s tariff is a cost-based tariff of the form (22).
One intuitive way to see why the tariff (22) is an equilibrium when θ is private information—
which is the basis of the proof of the result—is that this tariff would be the equilibrium if θ was
observable by firms, provided that the distribution of θ and x is independent. If a given consumer is
known to have taste parameters θ but unknown location x, then, by independence, this location is
uniformly distributed on [0, 1] as in the duopoly Hotelling model developed in Section 2, in which
we showed that for these type-θ consumers, firms would offer the two-part tariff (22), provided
the type-θ market is covered (which is ensured by (38)). Since this tariff does not depend on θ , it
is also an equilibrium if θ is private information to consumers.
In sum, provided that the market is sufficiently competitive so that (38) holds, one equilibrium
is given by each firm setting the simple cost-based tariff (22). Since this tariff achieves the firstbest outcome, allowing firms to offer general multiproduct nonlinear tariffs is clearly superior
in welfare terms to placing any form of restriction on the tariffs permitted (provided that this
equilibrium is chosen). This is a strong result, and it shows how competition can have the effect
of dramatically simplifying the tariffs offered by firms with market power. The simplicity of this
result contrasts strongly with the complex monopoly analysis in Armstrong (1996) and Rochet
and Choné (1998). In particular, as discussed in the Introduction, the monopoly model had the
feature that the firm would choose to exclude some consumers from its products, a feature that
made computation of the optimal nonlinear tariff difficult. With competition, a firm no longer has
the ability to exclude consumers, and this, among other factors, greatly simplifies the analysis.
A similar result has also been obtained by Rochet and Stole (forthcoming).30 They study
the problem under the assumptions that firms offer a single product and consumers have singledimensional private information θ that enters into utility functions so as to satisfy the singlecrossing property, and they obtain the parallel result. Importantly, they show that it is not important
that consumers be symmetrically distributed between the two firms. Obviously, in that case the
two firms offer different tariffs in equilibrium, but these tariffs are still cost-based two-part tariffs.
(Only the fixed charge differs across the firms.) The advantage of their more restrictive specification
is that they are able to find equilibria when the strong assumptions needed for Proposition 5 are
relaxed. In particular, they show that the result is highly sensitive to the assumptions that θ and
x are distributed independently and that all consumer types are willing to participate with the
candidate tariff. When the market is not fully covered, i.e., when (38) is violated, they show that
the market is segmented into two classes: one where consumers have strong tastes and where
competition is intense enough to drive marginal prices down to marginal costs, and one with
low-demand consumers over whom firms have a local monopoly.
30 For related analyses of screening models in duopoly, see also Desai (2001), who has a binary distribution for
the “vertical” taste parameter and who obtains a similar efficiency result, and Schmidt-Mohr and Villas-Boas (1997).
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5. Conclusions
This article has had two main themes. The first was to show some advantages of analyzing
firms as competitors in utility space, i.e., as simply supplying utility to consumers. Using the
basic relationship π(u) between profit and utility per consumer, we obtained directly—without
needing to examine costs, prices, and outputs—various results about the utility, profit, and welfare
effects of price discrimination. The second main theme was that freedom to engage in price
discrimination tends to be desirable in sufficiently competitive conditions: constraining the ways
that competitive firms supply utility to consumers usually reduces total welfare. (However, we
also saw that consumers often suffered when firms were granted extra freedom to extract surplus.)
The main exception to this rule was when consumers were segmented into identifiably separate
markets—or, alternatively, when consumers buy some products from one firm and some from
another, so that “one-stop shopping” is not the relevant framework—and when elasticities go the
“wrong” way.
Appendix
The proofs for Propositions 1, 4, and 5, Lemmas 1–4, and Corollary 1 follow.
Proof of Proposition 1.
Claim. The function (5) does have a maximum.
Proof. From (3), the function tends to −∞ as u → ū. Next, notice that the function
{log s(u, u) + log π (u)} − {φ(u) + log π (u)}
is decreasing, as its derivative is s2 /s ≤ 0. From (2) we know that the first term {·} is greater at u ∗ than at any
u < u ∗ . Since the above function is decreasing, we deduce that the same is true for the second term {·}, i.e., that
φ(u ∗ ) + log π (u ∗ ) ≥ φ(u) + log π (u) whenever u < u ∗ . Therefore, we know that there are points in the compact set
[u ∗ , ū −ε], for small ε > 0, that generate a higher value for (5) than any choice for u outside the set. Since (5) is continuous
in this set, we deduce that a maximum exists. Moveover, since the derivative of (5) is positive at u ∗ , this maximizer is
Q.E.D.
strictly greater than the collusive utility level u ∗ .
Therefore, let û be a utility level that maximizes (5). Suppose B chooses the utility û. If A chooses utility u A , the
log of its total profit is
log π (u A ) + log s(u A , û) = {log π (u A ) + φ(u A )} + {log s(u A , û) − φ(u A )} .
The first term {·} is maximized by assumption at u A = û and the second term {·} has derivative
s1 (u A , û)
s1 (u A , u A )
−
.
s(u A , û)
s(u A , u A )
Clearly this derivative is zero when û = u A , and from (1) it is positive when u A < û and negative when u A > û. In other
words, the second term {·} is single-peaked in u A and maximized at u A = û. We deduce that û is a best response for firm
A, and hence that û is a symmetric equilibrium.
Because the game is symmetric and involves upward-sloping reaction functions, there are no asymmetric equilibria.
Finally, suppose u is another symmetric equilibrium. The fact that u is A’s best response when B plays u implies that it
satisfies the first-order condition
φ (u ) +
π (u )
= 0.
π (u )
Therefore, if the function (5) is strictly concave, û is the unique maximizer of (5).
Q.E.D.
Proof of Lemma 1. This proof follows the method used for Proposition 1. Suppose that firm B offers utility û that
maximizes (19). If A chooses u A , the log of its profit is
uA
uA
log π (u A ) + log s(u A , û) = log π (u A ) +
+ log s(u A , û) −
,
t
t
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where s is given by (17). By construction, u A = û maximizes the first term {·} above, while the second term {·} has
derivative given by
1
1
−
t + u A − û
t
(A1)
1
1
− .
uA
t
(A2)
if t < u A + û, and otherwise is equal to
We claim that the derivative of the second term {·} is positive if u A < û and negative if u A > û, which implies that
u A = û is the best response for firm A. When t < u A + û, (39) is positive if u A < û and otherwise is negative. When
t > u A + û, it follows that u A < t, and so from (40) the derivative of the second term {·} is positive. All that remains to
show is that it is not possible that u A > û, that t > u A + û, and for t to satisfy (18). But since log π is concave, (18) implies
that the maximizer of log π(u) + u/t is no less than the maximizer of log π (u) + u/(2β), the latter by construction being
β. We deduce that û ≥ β. But (18) together with t > u A + û and û ≥ β implies that u A < û as required.
Q.E.D.
Proof of Lemma 2. From Lemma 1 we see that for sufficiently small t, û(t) satisfies the first-order condition
tπ (û(t)) + π (û(t)) = 0,
which can be differentiated to give
π + tπ + π û (t) = 0
(where the dependence of π on û(t) has been suppressed). This implies that û (0) = −1. Differentiating again gives
π û + tπ + π û + π + tπ û + π û û = 0,
which implies û (0) = π (ū)/π (ū). This gives the second-order Taylor expansion for û.
The other Taylor expansions follow from the definitions for π̂ and ŵ.
Q.E.D.
Proof of Lemma 3. Total differentiation of (24) yields
[2π π + τ 2 π ]û + 2τ π = 0,
and differentiating again gives
[2π π + τ 2 π ]û + [2π π + 2π π + τ 2 π ](û )2 + 4τ π û + 2π = 0.
Expression (24) then implies that
π (û(τ ))
→ −π (ū) as τ → 0.
τ
Letting τ → 0 in (A3) implies that
û (0) = − √
1
.
−π (ū)
Using these facts, dividing both sides of (A3) by 2τ , and letting τ → 0, we obtain that
2 2
3/2 2
π (û(τ )) û (τ ) + π (û(τ ))
π (ū)
û + π (ū) −π (ū) û (0) + 2π (ū)û (0) + lim
= 0.
τ →0
τ
Using L’Hôpital’s rule to evaluate the final term on the left-hand side of this expression, we have
2
3/2 û + π −π û + 2π û − π û + 2(π )2 û û = 0,
π
which, after substituting for the value for û , implies
û (0) = 0.
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From (25) we see that ŵ(0) = ū. Differentiating (25) we get
1
π π ŵ (τ ) = 1 +
û (τ )
1− 2
4
(π )
and so
5
1
ŵ (0) = − √
,
4 −π (ū)
which give the first two terms in the Taylor expansion. Differentiating again, and using the fact that û (0) = 0, gives
ŵ (0) =
which completes the proof.
π (ū)
,
4(π (ū))2
Q.E.D.
Proof of Lemma 4. From (5), let u i maximize of log πi (u) + φ(u). Then by revealed preference
log π D (u D ) + φ(u D ) ≥ log π D (u N ) + φ(u N )
log π N (u N ) + φ(u N ) ≥ log π N (u D ) + φ(u D ),
which implies that
log π D (u D ) − log π N (u D ) ≥ log π D (u N ) − log π N (u N ).
Hence from the assumption we have u D ≤ u N as required.
Q.E.D.
Proof of Corollary 1. From (15) we have
π D (u)
1
=−
.
π D (u)
v(c) − u
Since π N in (9) is concave,
0 = π N (v(c)) ≤ π N (u) + (v(c) − u)π N (u),
and so
π N (u)
π N (u)
≥−
π (u)
1
= D .
v(c) − u
π D (u)
This implies that log π D − log π N is decreasing in u, and we can apply Lemma 4 to this pair of profit functions and deduce
that consumer utility falls if two-part tariffs are used. Q.E.D.
Proof of Proposition 4. (i) Comparing (31) with (35), and (33) with (36), we see that this is true provided that
q̄ 2
< α + γ (1 − α),
φ
(A4)
which from the definition of q̄ and φ is true if and only if
γ q12 +
1 2
q2 2
√
γ q1 − √
q2 − 2q1 q2 =
γ
γ
is strictly positive, which is true unless q2 (c) = γ q1 (c).
(ii) Using the argument developed by Varian (1989)—see the Introduction to this article—we know that welfare is
lower with discrimination if total output is strictly lower with discrimination. Comparing (34) with (37), we see that this
is true in competitive markets provided that
q̄
q̄ 2 αq1 η1 + (1 − α)q2 η2
αη1 + γ (1 − α)η2 > −c q̄ =
.
φ
φ αq1 + (1 − α)q2
But from (A4) we see this is true provided that
© RAND
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αη1 + γ (1 − α)η2
αq1 η1 + (1 − α)q2 η2
,
≥
α + γ (1 − α)
αq1 + (1 − α)q2
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which, given (32), holds if and only if η2 ≤ η1 as claimed.
Q.E.D.
Proof of Proposition 5. Suppose B offers the tariff (22), so that u B (θ ) = v(θ ) − k − t. Given this tariff, an upper bound on
the available profit for firm A is obtained by supposing that it can observe a consumer’s parameter θ (but not her location
x). Therefore, suppose this is so, and for a consumer of type θ suppose A offers the gross utility u(θ ). The independence
assumption implies that the distribution of x is uniform on [0, 1] given θ , so from (17) the number of type-θ consumers
buying from A is
u(θ ) − v(θ ) − k − t
1
u(θ )
min
1+
,
.
2
t
t
The firm will choose the method of generating utility u(θ ) that maximizes its profits, which is done by setting prices
equal to costs, and setting the fixed fee f A (θ ) = v(θ ) − u(θ ). Therefore, A’s maximum profit available from the type-θ
consumers is
1
k + t − fA
v(θ ) − f A
max : ( f A − k) × min
1+
,
.
fA
2
t
t
When t is large the solution to this problem depends on θ via v(θ ). However, the problem has the solution f A (θ ) = k + t
provided that t ≤ (2/3)[v(θ ) − k]. In particular, provided that t is small enough to satisfy (38), A’s choice of tariff, given
that it can observe a consumer’s type θ , takes the form (22) for all θ . Since this tariff does not depend on θ , it must be the
best response for A when θ is private information.
Q.E.D.
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